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Just two steps to replicate change in name on remote as well as on GitHub:

Step 1 git branch -m old_branchname new_branchname

Step 2 git push origin :old_branchname new_branchname

EDIT Now mostly on servers like codecommit etc. step 2 can be replaced by

Step 2 git push --set-upstream origin new_branchname

Just three steps to replicate change in name on remote as well as on GitHub:

Step 1 git branch -m old_branchname new_branchname

Step 2 git push origin :old_branchname new_branchname

Step 3 git push --set-upstream origin new_branchname

Just two steps to replicate change in name on remote as well as on GitHub:

Step 1 git branch -m old_branchname new_branchname

Step 2 git push origin :old_branchname new_branchname

Just two steps to replicate change in name on remote as well as on GitHub:

Step 1 git branch -m old_branchname new_branchname

Step 2 git push origin :old_branchname new_branchname

EDIT Now mostly on servers like codecommit etc. step 2 can be replaced by

Step 2 git push --set-upstream origin new_branchname

Just two steps to replicate on remote as well as on GitHub:

Step 1 git branch -m old_branchname new_branchname

Step 2 git push origin :old_branchname new_branchname

Just two steps to replicate change in name on remote as well as on GitHub:

Step 1 git branch -m old_branchname new_branchname

Step 2 git push origin :old_branchname new_branchname